Complexity

Volume 2017 (2017), Article ID 1891923, 11 pages

https://doi.org/10.1155/2017/1891923

## Interval-Valued Intuitionistic Fuzzy Ordered Weighted Cosine Similarity Measure and Its Application in Investment Decision-Making

^{1}School of Business, Central South University, Changsha, China^{2}Department of Mathematics, Hunan University of Science and Technology, Xiangtan, China

Correspondence should be addressed to Donghai Liu

Received 13 October 2016; Revised 23 December 2016; Accepted 11 January 2017; Published 6 February 2017

Academic Editor: Jia Hao

Copyright © 2017 Donghai Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present the interval-valued intuitionistic fuzzy ordered weighted cosine similarity (IVIFOWCS) measure in this paper, which combines the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized ordered weighted averaging operator. The main advantage of the IVIFOWCS measure provides a parameterized family of similarity measures, and the decision maker can use the IVIFOWCS measure to consider a lot of possibilities and select the aggregation operator in accordance with his interests. We have studied some of its main properties and particular cases such as the interval-valued intuitionistic fuzzy ordered weighted arithmetic cosine similarity (IVIFOWACS) measure and the interval-valued intuitionistic fuzzy maximum cosine similarity (IVIFMAXCS) measure. The IVIFOWCS measure not only is a generalization of some similarity measure, but also it can deal with the correlation of different decision matrices for interval-valued intuitionistic fuzzy values. Furthermore, we present an application of IVIFOWCS measure to the group decision-making problem. Finally the existing similarity measures are compared with the IVIFOWCS measure by an illustrative example.

#### 1. Introduction

The similarity measure is an important tool for measuring the degree of similarity between two objects, which is very useful in some areas, such as decision-making, machine learning, pattern recognition, and medical diagnosis [1–6]. Over the past several decades, a variety of similarity measures have been introduced and investigated [7–14] based on intuitionistic fuzzy sets (IFSs) [15]. For example, Li and Cheng [9] investigated similarity measures on IFSs and showed how these measures may be used in pattern recognition problems. Later, Liang and Shi [10] introduced several new similarity measures on IFSs and discussed the relationships between these measures. Hung and Yang [11] presented a method to calculate the distance between IFSs based on the Hausdorff distance and used this distance to generate several similarity measures between IFSs. Furthermore, Hung and Yang [12] presented two new similarity measures between IFSs, which have been found to satisfy some similarity measure axioms. One of many similarity measures is the cosine similarity measure based on Bhattacharyya’s distance [13], which is defined as the inner product of two vectors divided by the product of their lengths. Ye [14] proposed a cosine similarity measure between IFSs and applied it to medical diagnosis and pattern recognition. However, in some cases, the degrees of membership or nonmembership are sometimes assumed not exactly as a number but as a whole interval; Atanassov and Gargov [16] introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs). Furthermore, Xu [17] developed some similarity measures of intuitionistic fuzzy sets and applied them to pattern recognition. Ye [18] proposed a cosine similarity measure for IVIFSs and applied it to multiple attribute decision-making problems.

When similarity measures are widely used in decision-making problems, the importance of ordered position of each degree of similarity should be emphasized. In other words, the higher the degree of similarity, the higher the weight which should be assigned to it; a very useful technique is the ordered weighted averaging (OWA) operator. The OWA operator is introduced by Yager [19], which is a very well-known aggregation operator that provides a parameterized family of aggregation operators including the maximum, the minimum, and the average as special cases. The prominent characteristic of the OWA operator is the reordering step. Since it has appeared, the OWA operator has been widely extended to other aggregation environments, including linguistic environment (Merigó and Casanovas [20], Wei and Zhao [21], and Zhou and Chen [22, 23]), fuzzy environment (Merigó and Gil-Lafuente [24], Xu [25]), intuitionistic fuzzy environment (Li [26], Zeng and Su [27], and Zhou et al. [28, 29]), and interval-valued intuitionistic fuzzy environment (Li et al. [30], Yu et al. [31], and Zhou et al. [32]), and used in areas such as decision-making and neural networks (Yager [33], Merigó and Gil-Lafuente [34], and Zhou et al. [35–38]).

The aim of this paper is to introduce the interval-valued intuitionistic fuzzy ordered weighted cosine similarity (IVIFOWCS) measure. It combines the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized OWA operator. A more complete formulation of the cosine similarity measure is obtained because it can consider parameterized families of operators that include the maximum, the minimum, and the average as special cases. Using the advantage of IVIFOWCS measure can relieve the influence of unduly large or unduly small deviations on the aggregation results. This measure provides a robust formulation that includes a wide range of particular cases, such as the interval-valued intuitionistic fuzzy ordered weighted arithmetic cosine similarity (IVIFOWACS) measure, the interval-valued intuitionistic fuzzy ordered weighted quadratic cosine similarity (IVIFOWQCS) measure, the interval-valued intuitionistic fuzzy ordered weighted geometric cosine similarity (IVIFOWGCS) measure, the interval-valued intuitionistic fuzzy maximum cosine similarity (IVIFMAXCS) measure, the interval-valued intuitionistic fuzzy minimum cosine similarity (IVIFMINCS) measure, the interval-valued intuitionistic fuzzy normalized cosine similarity (IVIFNCS) measure, the interval-valued intuitionistic fuzzy normalized arithmetic cosine similarity (IVIFNACS) measure, and the interval-valued intuitionistic fuzzy normalized geometric cosine similarity (IVIFNGCS) measure. The decision maker is able to consider a wide range of scenarios and select the one that is in accordance with his interests.

The paper is organized as follows. In Section 2, we briefly review the concepts of IFSs, IVIFSs, the cosine similarity measure for IVIFSs, and the OWA operator. In Section 3, we introduce the IVIFOWCS measure; some properties and different families of the IVIFOWCS measures are analyzed. Section 4 develops an application in the group decision-making problem. Section 5 gives a numerical example. Section 6 summarizes the main conclusions of the paper.

#### 2. Preliminaries

##### 2.1. Basic Concepts of IFSs and IVIFSs

*Definition 1. *Let be a finite universal set; IFs in is defined aswhere are the membership function and nonmembership function, respectively, such that .

Assume ; then is called the hesitation degree of whether belongs to or not. It is obvious that . For convenience, we call an intuitionistic fuzzy number (IFN) and denote the module of as .

*Definition 2. *Let ; IVIFs in is defined as , where intervals and denote the membership degree and nonmembership degree of the element to the set , respectively. For each , the hesitancy degree of an interval intuitionistic fuzzy set is defined as follows: An interval-valued intuitionistic fuzzy number (IVIFN) ; we denote the module of as .

Let and be two IVIFNs; the operations are defined as follows (Ye [18]):(1)(2)(3) if and .

##### 2.2. The OWA Operator

The OWA operator is an aggregation operator that provides a parameterized family of aggregation operators that includes the maximum, the minimum, and the average as special cases. It can be defined as follows.

*Definition 3. *An OWA operator of dimension is a mapping OWA: that has an associated weighting vector with and , such that , where is the largest th of the arguments

Note that the OWA operator is commutative, monotonic, bounded, and idempotent.

Yager [39] developed the generalized OWA (GOWA) operator, which is defined as follows.

*Definition 4. *A GOWA operator is a mapping GOWA: that has an associated weighting with and , and a parameter and , such that where is the largest th of the arguments .

We know that the GOWA operator is also commutative, monotonic, bounded, and idempotent (Yager [39]). We can obtain a group of particular cases. For example, if , then the GOWA operator is reduced to the OWA operator. If , the ordered weighted geometric averaging (OWGA) operator is obtained. If , the ordered weighted harmonic averaging (OWHA) operator is formed.

##### 2.3. Cosine Similarity Measures for IVIFSs

*Definition 5. *Let , assume that there are two IVIFSs and , and a cosine similarity measure between two IVIFSs and is defined as follows:

The cosine similarity measure between and satisfies the following properties:(1)(2)(3) if , i.e., , , and . .

#### 3. Interval-Valued Intuitionistic Fuzzy Ordered Weighted Cosine Similarity Measure

In this section, we will introduce the IVIFOWCS measure, which is a similarity measure that uses the cosine similarity measure for IVIFS in the GOWA operator.

##### 3.1. The IVIFOWCS Measure

Let , be two interval-valued intuitionistic fuzzy matrices, , are IVIFNs for all , and assume that and for . We can define the IVIFOWCS measure as follows.

*Definition 6. *An IVIFOWCS measure of dimension is a mapping IVIFOWCS: that has an associated weighting vector with and , such that where is the cosine similarity measure between IVIFS and and is any permutation of , such that

*Remark 7. *If in and , the IVIFOWCS measure reduces to the cosine similarity measure for IVIFS (Ye [18]).

*Example 8. *LetBy (4), we can get , , and .

Then , , and .

If , by using the IVIFOWCS measure, we can obtain the cosine similarity measures corresponding to some special cases of the parameter , which are shown in Table 1.